Trigonometry
Grade 6-8The Cosine Rule
The cosine rule handles the two triangle situations the sine rule cannot: when two sides and the angle between them are known, or when all three sides are known but no angle. This lesson covers the cosine rule formula for finding a side, the rearranged formula for finding an angle, choosing between the sine rule and the cosine rule, and finding the largest angle in a triangle.
What you need to know
- Use the cosine rule for three sides, or for two sides and the angle between them.
- a squared equals b squared plus c squared minus 2 b c cos A to find a side.
- Rearranged, cos A equals (b squared plus c squared minus a squared) over 2 b c to find an angle.
- If the angle is 90 degrees, cos A is 0 and the rule reduces to Pythagoras.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
The cosine rule formula for finding a side
When two sides and the angle between them are known (SAS), use a² = b² + c² - 2bc cos A, where a is the side opposite angle A, and b and c are the two known sides either side of that angle.
With b = 8 cm, c = 10 cm, and the included angle A = 60°: a² = 8² + 10² - 2 × 8 × 10 × cos 60° = 84, so a = √84 = 9.17 cm (3 s.f.).
The cosine rule formula for finding an angle
When all three sides are known but no angle (SSS), rearrange the formula to make cos A the subject: cos A = (b² + c² - a²) ÷ (2bc). Substitute the three sides, then use cos⁻¹ to find the angle.
With a = 7 cm, b = 9 cm, c = 8 cm: cos A = (8² + 9² - 7²) ÷ (2 × 8 × 9) = 0.667, so A = cos⁻¹(0.667) = 48.2° (1 d.p.).
Choosing between the sine rule and the cosine rule
Use the cosine rule when two sides and the included angle are known (SAS), or when all three sides are known (SSS). Use the sine rule instead when a matching angle-side pair is known, alongside one more angle or side.
Given all three sides of a triangle with no angle known, the cosine rule must be used, since the sine rule requires a known angle-side pair to start from.
Finding the largest angle in a triangle
The largest angle in a triangle is always opposite the longest side. When asked to find the largest angle without being told which one it is, first identify the longest side, then use the cosine rule to find the angle directly opposite it. A negative value for cosine simply means the angle is obtuse, between 90° and 180°.
In a triangle with sides 6 cm, 8 cm and 11 cm, the longest side is 11 cm, so the largest angle is opposite it, and the cosine rule gives that angle as 102.6° (1 d.p.).
Worked examples
Three exam-style questions, fully solved
In triangle ABC, side b = 10 cm, side c = 8 cm, and the included angle A = 60°. Find the length of side a. Give your answer correct to 3 significant figures.
Easy- 1.Substitute into the formula: a² = 8² + 10² - 2 × 8 × 10 × cos 60°
- 2.Calculate: a² = 64 + 100 - 80 = 84
- 3.Take the square root: a = √84
Answer: 9.17 cm
In triangle ABC, side a = 7 cm, side b = 9 cm, and side c = 8 cm. Find the size of angle A. Give your answer correct to 1 decimal place.
Medium- 1.Substitute into the rearranged formula: cos A = (8² + 9² - 7²) ÷ (2 × 8 × 9)
- 2.Calculate: cos A = 96 ÷ 144 = 0.667
- 3.Use the inverse cosine function: A = cos⁻¹(0.667)
Answer: 48.2°
Triangle ABC has sides AB = 8 cm, BC = 6 cm and CA = 11 cm. Find the size of the largest angle in the triangle. Give your answer correct to 1 decimal place.
Hard- 1.Identify the longest side, CA = 11 cm, opposite angle B, so B is the largest angle
- 2.Substitute into the formula: cos B = (6² + 8² - 11²) ÷ (2 × 6 × 8)
- 3.Calculate and use the inverse cosine function: cos B = -0.219, so B = cos⁻¹(-0.219)
Answer: 102.6°
Practice
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Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
In triangle ABC, b = 7 cm, c = 9 cm, and angle A = 50°. Work out the length of side a, to 1 decimal place.
Work out the length of the side marked x, to 1 decimal place.
In triangle ABC, a = 6 cm, b = 8 cm and c = 12 cm. Work out the size of angle C, to 1 decimal place.
Work out the size of the angle marked x, to 1 decimal place.
Two roads leave a junction at an angle of 70°. One car drives 4 km along one road and another car drives 6 km along the other. How far apart are the two cars, to 1 decimal place?
In triangle ABC, a = 9 cm, b = 10 cm and angle C = 45°. Work out the length of side c, to 1 decimal place.
Avoid these
The mistakes students make most often
Confusing which side is being found and which two sides go into the formula, substituting the wrong side as the subject.
Making an arithmetic slip with the negative term in the formula, especially when squaring and subtracting several values in one step.
Treating a negative cosine value as an error rather than recognising it correctly gives an obtuse angle between 90° and 180°.
Using the cosine rule when the sine rule is actually needed, such as when two angles and a side are known.
Forgetting that the largest angle in a triangle is always opposite the longest side when asked to find it without more detail.
FAQ
Questions students and parents ask
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